Interactions and superconductivity in heavily doped MoS2

0.59 MB·English

by  R. Roldan

Interactions and superconductivity in heavily doped MoS
2
R. Rold´an1, E. Cappelluti1,2, F. Guinea1
1 Instituto de Ciencia de Materiales de Madrid. Consejo Superior de
Investigaciones Cient´ıficas. Sor Juana In´es de la Cruz 3, 28049 Madrid. Spain.
2 Istituto de Sistemi Complessi, U.O.S. Sapienza, CNR, v. dei Taurini 19, 00185 Roma, Italy
Weanalyzethemicroscopicoriginandthephysicalpropertisofthesuperconductingphaserecently
observed in MoS . We show how the combination of the valley structure of the conduction band,
2
the density dependence of the screening of the long range Coulomb interactions, the short range
electronic repulsion, and the relative weakness of the electron-phonon interactions, makes possible
the existence of a phase where the superconducting order parameter has opposite signs in different
3
valleys, resembling the superconductivity found in the pnictides and cuprates.
1
0
PACSnumbers: 74.20.-z,74.20.Mn,74.70.-b
2

n
a Molybdenum disulfide (MoS ) is a layered semicon-
J 2 a b
ductor which can be exfoliated down to monolayer unit ~q
�
1
cells [1], like graphene [2–4]. The existence of an en-
2 ~q
ergy gap makes MoS a convenient material for nano-
2
] electronics [5, 6]. Metallic behavior can be induced, also K’ � Q K
n
like in graphene, by means of electric field effects or by
o
doping, and the corresponding Fermi surface is typically ~q
c �
- made up by inequivalent Fermi pockets [7–11], defining ~q
r
a valley degree of freedom which is strongly entangled
p
u with the spin degree of freedom [12], and it can be fur- c d
K K’ K’ K
s thercontrolledandmanipulated,openingpromisingper-
.
t spectives for spintronics. At high carrier concentrations
ma (n 1014cm−2),andinthepresenceofhigh-κdielectrics, K Vintra K’ K Vinter K’
∼
MoS has also be shown to undergo a superconducting
- 2
d transition, withadoping-dependentcriticaltemperature
n T (n) which exhibits a maximum as function of n and e f -Δ +Δ
c
o drops to zero at sufficiently large values of n [13, 14]. +Δ +Δ -Δ -Δ
c +Δ -Δ -Δ +Δ
[ A ferromagnetic behavior has been also reported in
MoS2 [15–18], and it has been related to edges or to the +Δ -Δ
1 existence of defects [19, 20]. The magnetic properties -Δ +Δ -Δ -Δ +Δ +Δ
v
+Δ -Δ
of MoS nanoribbons indicate that the electron-electron
6 2
3 interactions are non negligible. The combination of sig- +Δ -Δ -Δ +Δ
8 nificant electron-electron interactions and a two dimen- +Δ +Δ -Δ -Δ
4 sional Fermi surface made up of many pockets is also -Δ +Δ
.
1 a hallmark of the cuprate and pnictide superconductors
0 [21], where the superconducting gap has a d-wave sym- FIG. 1: Sketch of the intravalley (a) and intervalley (b)
3 metry (cuprates) or opposite sign in different pieces of scattering processes between Cooper pairs in MoS2. Pan-
1 els (c) and (d): corresponding Feynman diagrams associated
the Fermi surfaces (pnictides) [21]. A related gap struc-
: to intra- and intervalley scattering. Full arrows are electron
v ture has been also proposed for heavily doped graphene
propagators, and wavy lines are effective interactions. The
i
X whentheelectron-electroninteractionissufficientlylarge possible unconventional superconducting phases discussed in
[22]. the text are sketched in panels (e) and (f).
r
a In the present work we study the origin of supercon-
ductivity in heavily doped MoS , by considering the role
2
of both electron-electron and electron-phonon interac-
tions. We analyze first the general features of the ef- determination of the superconducting Tc is outside the
fective interaction between charge carriers, and we make scope of our work, the present analysis suggests that su-
semi-quantitative estimates of the strength of the differ- perconductivity in MoS2 is likely to be induced by the
ent contributions to the effective coupling. We discuss electron-electron interaction, and that a superconduct-
next the competition between the electron-electron and ing phase with a non trivial gap structure is possible.
electron-phonon interactions, and the possible types of Effective interactions. Following the experimental re-
superconductivity that emerge. Although a quantitative sultsofRef. [13,14],weassumethatthecarriersleading
2
to the superconducting phase are electron like, confined [25], finding relevant contributions from three acoustic
in the first MoS layer closer to the high-dielectric gate. modes and from six optical modes. Note that the cou-
2
Thevalidityofthisapproximationwillbediscussedlater. pling to the acoustic modes vanishes when the phonon
We also assume that, similar to monolayer MoS , the wavevectorapproacheszero. Theleadingcouplingsiden-
2
electron-like carriers are located in the two inequivalent tified in Ref. [25] are thus the ones to the polar LO
Fermi pockets centered at the K and K(cid:48) corners of the modes, (cid:126)ω 0.048 eV, and to the homopolar mode,
LO
BrillouinZone(seeFig. 1),havingthusasizabled-orbital (cid:126)ω 0.05 e≈V, where the S atoms oscillate out of the
ho
≈
character with main d orbital component. At suf- plane [25]. The homopolar mode contributes to intraval-
3z2−r2
ficiently large concentrations six additional inequivalent leyscattering,whiletheLOmodecontributestointraval-
Q valleys start to be filled, located half way between the ley scattering, through the induced electric polarization,
Γ and K point, with primary d and d Mo orbital and also to intervalley scattering.
x2−y2 xy
character. Thesesecondaryvalleysathigherchargeden- Using the notation in [25], we define an effective in-
sity do not play a relevant role in our main discussion teraction, at frequencies much smaller than the phonon
andwillbethereforeneglected. Theirpossibleeffectwill energies, as
be however addressed in the final discussion.
We explore here the possibility that superconductiv- VLO = gL2O Ω,,
ity is induced by effective electron-electron interactions, intra −(cid:126)ω ×
LO
associated with the direct Coulomb interaction between D2 (cid:126)
VLO = LO Ω,
charge carriers or with the effective coupling induced by inter −(cid:126)ω 2Mω ×
LO LO
phonons. As in Ref. [22], we consider singlet super- D2 (cid:126)
conductivity. Because of the time invariance symmetry, Vho = ho Ω, (2)
inter −(cid:126)ω 2Mω ×
the Cooper pair electrons (k , k ) reside in differ- ho ho
↑ − ↓
ent valleys. We can now classify, as sketched in Fig. 1, where g 0.098 eV is the long wavelength polar cou-
LO
theinteractionsleadingtoscatteringoftheCooperpairs pling, D ≈ 2.6 eV ˚A−1 and D 4.1 eV ˚A−1 are de-
LO ho
intointra-andintervalleycouplings,namelyVintra((cid:126)q,ω), formation p≈otentials for the LO an≈d homopolar modes,
Vinter((cid:126)q,ω), where (cid:126)q and ω are the exchanged momen- respectively, and Ω is the area of the unit cell. We take
tum and frequency. We consider only scattering pro- forM themassofthesulfuratom, whichismuchlighter
cesses of carriers near the Fermi surfaces, which are as- thanMo,andthusexpectedtobedominant. Addingthe
sumedtobeisotropicandcenteredatK andK(cid:48), andwe three contributions, we find thus:
neglect the frequency dependence of the interaction.
The classification of the interaction in an intravalley λph 0.36,
intra ≈−
and an intervalley component allows us to define the di-
λph 0.13, (3)
mensionless coupling constants [22] inter ≈−
(cid:90) π (cid:20) (cid:18)θ(cid:19)(cid:21) whichaccountfortherespectiveintravalleyandinterval-
λ =ρ((cid:15) ) dθ V 2k sin , (1)
α F α F ley electron-phonon coupling constants.
2
0
Electron-electron interaction. After having estimated
whereαlabelstheintra-andinter-valleyscattering,and
the electron-phonon coupling, we address now the
ρ((cid:15) )=m /(2π(cid:126)2)isthedensityofstatesattheFermi
F eff electron-electron repulsive interaction. Intravalley scat-
levelpervalleyandperspinintermsoftheeffectivemass
tering is operative only for small momenta/large dis-
m . Typical values are m 0.5m [23, 24], where
eff eff 0 tance, where, as in Ref. [22], the electron-electron inter-
≈
m is the free electron mass. For realistic charge con-
0 action is determined by the screened Coulomb potential:
centrations the Fermi wavevector is much smaller than
the dimensions of the Brillouin Zone, kF (cid:28)|K(cid:126)|, so that Ve−e (q)= 2πe2 , (4)
we can in good approximation neglect the momentum intra (cid:15) (q+q )
0 FT
dependence of V ((cid:126)q),V ((cid:126)q), except for electron-
intra inter
electron intravalley scattering, as discussed below. where (cid:15) is the dielectric constant of the environment,
0
The existence of superconductivity requires λ and q = 2πe2ρ((cid:15) )/(cid:15) is the Thomas-Fermi wave-
intra FT F 0
±
λ < 0, where the choice of the sign depends on vector.
inter
whetherthegapsinthetwovalleyshaveequaloropposite On the other hand, the contribution of the electron-
signs. The electron-phonon coupling leads to attractive electroninteractiontointervalleyscatteringisassociated
interactions, λe−ph < 0, while the electron-electron cou- withtheshortrangepartoftheCoulombpotential. The
plings lead to repulsive interactions, λe−e >0. However, leadingterminthisinteractionisthusgivenbytheHub-
repulsive interactions can also lead to superconductivity bard term, namely the repulsion between two electrons
provided that λ >λ >0. with opposite spin in the same atomic orbital. Since,
inter intra
Electron-phonon interaction. The electron-phonon in- as we discussed above, electronic states close to the K
teractioninsinglelayerMoS hasbeenevaluatedinRef. and K(cid:48) points have a dominant Mo 4d character, we can
2
3
Λintra,Λinter sion, but a more accurate determination of its specific
value is lacking. In this situation, we will consider U
1.0 4d
and (cid:15) as a free variable parameters and we will inves-
0
0.8 tigate their effects on the superconducting properties of
MoS in the range (cid:15) 10 50 and U 2 10 eV.
2 0 4d
0.6 ≈ ÷ ≈ ÷
Fig. 2 shows the dependence of the intravalley and in-
0.4 tervalley electron-electron coupling constants λein−tera and
λe−e on the specific values of (cid:15) and U . In particu-
inter 0 4d
0.2 lar, the intravalley coupling λe−e shows an initial sig-
intra
nificant dependence on density, due to the screening in-
ncm(cid:45)2
0 1(cid:180)1014 2(cid:180)1014 3(cid:180)1014 4(cid:180)1014 crease with density for qFT (cid:46) kF, wheeras λein−tera satu-
rates for q k . A comparison with the correspond-
TF F
(cid:29)
FIG. 2: Intravalley and intervalley dimensionless couplings ing coupling constants for electron-phonon interaction,
(cid:72) (cid:76)
λein−tera,λein−teerduetoelectron-electroninteractionsasfunctions provided by Eq. (3), shows that the values of λein−tera and
of density n, for different values of (cid:15)0 and U4d. Full lines: λe−e in the limit of high dielectric constant and mod-
λ . (from top to bottom: (cid:15) = 10,20,30,40,50); broken inter
intra 0 erate U are larger in this regime than those of λe−ph
4lineeVs:, 2λienVte)r.(from top to bottom: U4d =10 eV, 8 eV, 6 eV, andλe−4pdh,suggestingthusthatsuperconductivityisindturae
intra
to the electron-electron interaction. Furthermore, since
λ = λe−ph+λe−e > 0, the superconducting phase
inter inter inter
Ε0 is expected to have gaps with opposite signs in the two
50
valleys, as sketched in Fig. 1e. Note that superconduc-
tivity of this type is possible only when the carrier den-
40
sity n satisfies the condition λ (n) λ <0. This
intra inter
−
30 inequality defines a critical density nc above which su-
perconductivity appears. The dependence of n on the
c
20 dielectricconstantoftheenvironmentisshowninFig.3.
Apart from the more likely symmetry of the gap dis-
10
cussed above, it is worth to notice that an additional su-
0 n cm(cid:45)2 perconductingphaseispossibleduetothemodulationof
c
1(cid:180)1014 2(cid:180)1014 3(cid:180)1014 4(cid:180)1014 the electron-electron interaction due to screening. This
mechanism always leads to an order parameter with a
FIG. 3: Relation between the critical carrier de(cid:72)nsity,(cid:76)nc, modulated k-dependent gap within each valley [27] (see
required for the existence of superconductivity and the di-
Fig. 1f). However, the resulting critical temperatures
electric constant of the environment, (cid:15) . Solid blue curve:
0 for this phase are typically very low [22], so that it is
U =2 eV, dashed red curve: U =4 eV.
4d 4d
very unlike to be related to the experimental evidence of
superconductivity in this material.
Superconducting phase. The change in sign of the su-
therefore approximate:
perconducting gap in the two valleys implies that the
Ve−e U Ω, (5) superconducting phase in MoS2 have unusual proper-
inter ≈ 4d× ties with respect to conventional superconductors. More
resulting thus in a k-independent and density- specifically we can remind [27]: i) Elastic scattering is
independent interaction. pair breaking, leading to the suppression of supercon-
Results. We discuss now the consequences of the ductivitywhenvF/(cid:96)(cid:38) ∆,wherevF =(cid:126)kF/meff isthe
| |
electron-phonon vs. electron-electron and the intra- vs. Fermi velocity, (cid:96) is the elastic mean free path, and ∆ is
intervalley interaction in regards to the superconducting the superconducting gap; ii) Strong scatterers induce lo-
order. Along this line, unfortunately, the values of the calizedAndreevstateswithinthegap;iii)Andreevstates
dielectricconstant,(cid:15) ,andoftheHubbardterm,U ,are also appear at certain edges. However, unlike graphene,
0 4d
notwellknown. Theexperimentsreportedin[13,14]are itshouldberemindedtahtMoS2 alsoshowsasignificant
done in the presence of a dielectric with a high value of spin-orbitcoupling,sothatthecombinationofanontriv-
(cid:15) . The ionization energies [26] of the Mo atom allow us ial superconducting order parameter with the spin-orbit
0
tomakeanorderofmagnitudeestimateofU . Alterna- coupling can lead to other interesting properties.
4d
tively, we can calculate the Coulomb integral for the 4d Open questions. An intriguing role, in the above sce-
orbitalsofMo,orreplacetheorbitalbyachargedsphere nario,isplayedbythespecificvalueoftheeffectivemass
withthesameradius. Allthesedifferentapproachescon- m . Apartfromthedeterminationoftheeffectivecou-
eff
verge on the order of magnitude of the Hubbard repul- plings, such parameter is relevant in assessing the ro-
4
bustness of the Fermi surface structure depicted in Fig.
1. The previous analysis was based on the assumption
of an electronic structure similar to thant in monolayer
MoS ,wherethecarriersoccupytwoinequivalentvalleys
2
corresponding at the absolute minima in the conduction
band at the K and K(cid:48) points. However, it should be re-
minded that six inequivalent secondary minima are also
predictedattheQpoints,midwaybetweentheΓandthe
K andK(cid:48) points,shownbythedotsinFig. 1. Inbilayer
and multilayer MoS the minima at the Q points are ex-
2
pected to lie below the valleys at the K and K(cid:48) points,
andinmonolayerMoS theminimainQstarttobefilled
2
at sufficiently high carrier densities, once the pockets in
K and K(cid:48) are partially filled. Since the relative filling of
the pockets at K-K(cid:48) and Q points depends on the cor- FIG. 4: Distribution of the electric charge per layer n(i) in
responding effective masses, an accurate determination multilayer MoS2 under gated conditions for n = 1014 cm−2,
for different values of the effective mass.
of m could also assess the possible presence of Fermi
eff
pocketsattheQpoints. Itshouldbealsonotedthatthe
valueoftheeffectivemassm determinesthescreening
eff
properties and, as a result, the distribution of the gate-
induced charge in the multilayer MoS . In Ref. [14] it
2
was suggested that, because of the strong electric field,
As final consideration, it should be remarked that our
carriersaremainlylocalizedinthefirstlayerclosetothe
analysis implies that that the main dependence of the
high-dielectric gate (note that the high carrier densities
superconducting phase on the charge density is by the
enhance the screening and thus the confinement).
suppression of the long range Coulomb repulsion, due to
Theoretical first-principles-based calculations indicate
thehigherscreeningathigherchargeconcentrations(see
a mass m of the order m 0.4 1m [7–
eff eff ≈ − 0 Fig.2). Hence,ouranalysisatthisleveldoesnotaccount
11,23,24,28],whereasearlyexperimentalmeasurements
for the decrease of T observed at high carrier densities
suggest m ranging from 0.01m [29] to 1m [30– c
eff ∼ 0 ∼ 0 n(cid:38)1.3 1014cm−2 [14]. A possible explanation for this
32]. In the absence of a definitive estimate of meff, we behavior×could be the change in the orbital character
checked the dependence of the induced charge distribu-
of the states close to K and K(cid:48) point, which loose 4d
tion on m by performing a Thomas-Fermi-like calcu-
eff orbital character at higher doping, resulting in a weaker
lation using a similar model for the screening of a MoS
2 Hubbard-likeintervalleyinteraction,orthechangeinthe
multilayer as in Ref. [33]. The results for n 1014cm−2
≈ topology of the Fermi surface, as the Q valleys start to
using different possible values of m , are shown in
eff be filled.
Fig. 4. They confirm that for m (cid:38) 0.1m the charge
eff 0
is indeed confined only in the layer closest to the high-κ
gate [14]. Note that, since the charge is strongly concen- Conclusions. In this paper we have analyzed the ap-
trated in the first layer, an appropriate generalization of pearing of a superconducting phase of MoS2 at high car-
the model on a discrete layer distribution was here em- rier concentrations and for strong screening of the long
ployed,whereasthecontinuumoriginalmodelofRef. [33] rangeCoulombpotential. Thesignificantshortrangere-
wouldbeinappropriate. Ourresultscorroboratethusthe pulsion between carriers at the conduction band allows
initialassumptionthatthesinglelayermodelisvalidand for a superconducting phase induced by the electron-
only the K and K(cid:48) valleys are occupied for the densities electron interaction, where the gap acquires opposite
that lead to superconductivity in [13, 14]. signs in the two inequivalent pockets of the conduction
band. This superconducting state, similar to that found
It is worth to mention however that, exotic supercon-
inthecuprateandpnictidesuperconductors, isexpected
ductingphases,withagapwithdifferentsignsindifferent
toshowinterestingtopologicalfeatures,suchasAndreev
valleysassketchedinFig. 1e,areexpectedtoappearalso
states at edges and grain boundaries.
inthecaseswhenonlytheFermipocketsattheQpoints
are filled (expected in the low density regime when the
carriers are spread among a few layers) or even in the Acknowledgements. F. G. acknowledges financial sup-
case at higher fillings where both pockets (Q and K) are port from MINECO, Spain, through grant FIS2011-
occupied. Such exotic phases would share a similar phe- 23713, and the European Union, through grant 290846.
nomenology as in the case here considered of only two R.R.acknowledgesfinancialsupportfromtheJuandela
valleys at the K and K(cid:48) points, in particular in regards Cierva Program (MINECO, Spain). E.C. acknowledges
to the features (i)-(iii) mentioned above. the Marie Curie grant PIEF-GA-2009-251904.
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